[Greenbuilding] A Solar Hot Tub

[Nick Pine]

Dan writes:

> Your calculations are meaningless unless you specify what types of components you're using: panels, storage tank, type of heater, and system design.

Here's a complete thermal specification:

20 PI=4*ATN(1)
30 D=5'tub diameter (feet)
40 H=4'tub depth (feet)
50 ALID=PI*(D/2)^2'lid area (ft^2)
60 ATUB=PI*D*H-ALID'tub area, without lid (ft^2)
70 RV=15'R-value of tub and lid (ft^2-F-h/Btu)
80 PRINT"    Eon        Eoff          Esun          Elec          Delay"
90 PRINT"    (Btu/day)  (Btu/day)     (kWh/day)     (kWh/day)     (minutes)
110 '1. 100% solar heat
120 EON=22*(104-45.3)*(ALID+ATUB)/RV'lid-on energy (Btu/day)
130 PW=EXP(17.863-9621/(460+104))'tub vapor pressure ("Hg)
140 PA=29.921/(1+.62198/.0087)'air vapor pressure ("Hg)
150 EOFF=2*((104-45.3)*ATUB/RV+100*ALID*(PW-PA))'lid-off energy (Btu/day)
160 SKWH1=(EON+EOFF)/3412'solar heat (kWh/day)
170 PRINT 1;EON,EOFF,SKWH1,EKWH,DELAY
180 '2. solar heat for 6 hours per day
190 ESUN=6*(104-45.3)*(ALID+ATUB)/RV/3412'solar energy (kWh/day)
200 PRINT 2;EON,EOFF,ESUN,SKWH1-ESUN,DELAY
210 '3. solar heat with delay, sunny day
220 G=(ALID+ATUB)/RV'tub conductance (Btu/h-F)
230 C=ALID*H*62.33'thermal capacitance (Btu/F)
240 RC=C/G'time constant (hours)
250 T6=45.3+(104-45.3)*EXP(-3/RC)'tub temp at 6 PM (F)
260 REHEAT=(104-T6)*C/3412'reheat energy (kWh/day)
270 PHEAT=1.5'heater power (kW)
280 DELAY=60*REHEAT/PHEAT'reheat delay (minutes)
290 T9=45.3+(104-45.3)*EXP(-13/RC)'tub temp at 9 AM (F)
300 EON=(104-T9)*C+6*(104-45.3)*G+REHEAT*3412
310 ESUN=((104-T9)*C+6*(104-45.3)*G)/3412
320 ELEC=REHEAT+EOFF/3412
330 PRINT 3;EON,EOFF,ESUN,ELEC,DELAY
340 '4. solar heat with delay, cloudy day
350 T6=45.3+(104-45.3)*EXP(-22/RC)'tub temp at 6 PM (F)
360 REHEAT=(104-T6)*C/3412'reheat energy (kWh/day)
370 DELAY=60*REHEAT/PHEAT'reheat delay (minutes)
380 ESUN=0
390 ELEC=REHEAT+EOFF/3412
400 PRINT 4;REHEAT*3412,EOFF,ESUN,ELEC,DELAY
410 '5 solar heat with no electric reheat
420 T5D=104'initial tub temp (F)
430 FOR DAY=1 TO 5
440 EON=22*(T5D-45.3)*(ALID+ATUB)/RV'lid-on energy (Btu/day)
450 PW=EXP(17.863-9621/(460+T5D))'tub vapor pressure ("Hg)
460 EOFF=2*((T5D-45.3)*ATUB/RV+100*ALID*(PW-PA))'lid-off energy (Btu/day)
470 T5D=T5D-(EON+EOFF)/C'tub temp after 5 cloudy days (F)
490 NEXT DAY
500 DELAY=0
520 ELEC=0
530 PRINT 5;EON,EOFF,ESUN,ELEC,DELAY,T5D

    Eon        Eoff          Esun          Elec          Delay       Final temp
    (Btu/day)  (Btu/day)     (kWh/day)     (kWh/day)     (minutes)   (F)

 1  5409.405   7496.244      3.782429      0             0           104
 2  5409.405   7496.244      .4323834      3.350046      0           104
 3  5390.759   7496.244      1.364025      2.41294       8.636646    104 
 4  5358.812   7496.244      0             3.767602      62.82312    104
 5  4535.253   5219.712      0             0             0           92.52149

Perhaps you'd like to change something. More water would help a little in case 5, but there's already lots of water, about 600 gallons. More insulation won't help much, since the main heat loss happens with the cover off.

> Your "five cloudy day" scenario simply won't work unless you drastically oversize the system. 

I disagree (compared to what? :-) Evacuated tubes can make very hot water very efficiently.
 
> Have you ever had a hot tub or worked with one?

Not really, altho I helped raise a water temp over 104 by adding a resistor in series with a thermistor and helped design an ozone system for cryptosporidium control in our local YMCA's 3500 gallon verson, as well as their 380K gallon pool. And Gary and I are working on an inexpensive pond/tub-type solar water heater at http://www.builditsolar.com/Experimental/DHWPond/Proto3/SolarPondDHW.htm Did Sir Isaac Newton (1642-1747) own a hot tub? :-) He invented Newton's law of cooling (see above), as well as calculus and gravity. This is 300-year-old physics.

Nick